+---
+geometry: margin=1.9cm
+columns: 2
+graphics: yes
+author: Andrew Lorimer
+---
+
# Circular functions
+Period of $a\sin(bx)$ is ${2\pi} \over b$
+
+Period of $a\tan(nx)$ is $\pi \over n$
+Asymptotes at $x={2k+1)\pi \over 2n} \> \vert \> k \in \mathbb{Z}$
+
## Reciprocal functions
### Cosecant
+
+
$$\operatorname{cosec} \theta = {1 \over \sin \theta} \> \vert \> \sin \theta \ne 0$$
- **Domain** $= \mathbb{R} \setminus {n\pi : n \in \mathbb{Z}}$
- **Turning points** at $\theta = {{(2n + 1)\pi} \over 2} \> \vert \> n \in \mathbb{Z}$
- **Asymptotes** at $\theta = n\pi \> \vert \> n \in \mathbb{Z}$
-
### Secant
+
+
$$\operatorname{sec} \theta = {1 \over \cos \theta} \> \vert \> \cos \theta \ne 0$$
- **Domain** $= \mathbb{R} \setminus \{{{(2n + 1) \pi} \over 2 } : n \in \mathbb{Z}\}$
- **Turning points** at $\theta = n\pi \> \vert \> n \in \mathbb{Z}$
- **Asymptotes** at $\theta = {{(2n + 1) \pi} \over 2} \> \vert \> n \in \mathbb{Z}$
-
### Cotangent
+
+
$$\operatorname{cot} \theta = {{\cos \theta} \over {\sin \theta}} \> \vert \> \sin \theta \ne 0$$
- **Domain** $= \mathbb{R} \setminus \{n \pi: n \in \mathbb{Z}\}$
1 + \operatorname{cot}^2 x & = \operatorname{cosec}^2 x, \quad \text{where } \sin x \ne 0 \\
1 + \tan^2 x & = \operatorname{sec}^2 x, \quad \text{where } \cos x \ne 0
\end{split}\end{equation}
+
+## Compound angle formulas
+
+$$\cos(x \pm y) = \cos x + \cos y \mp \sin x \sin y$$
+$$\sin(x \pm y) = \sin x \cos y \pm \cos x \sin y$$
+$$\tan(x \pm y) = {{\tan x \pm \tan y} \over {1 \mp \tan x \tan y}}$$
+
+## Double angle formulas
+
+\begin{equation}\begin{split}
+ \cos 2x &= \cos^2 x - \sin^2 x \\
+ & = 1 - 2\sin^2 x \\
+ & = 2 \cos^2 x -1
+\end{split}\end{equation}
+
+$$\sin 2x = 2 \sin x \cos x$$
+
+$$\tan 2x = {{2 \tan x} \over {1 - \tan^2 x}}$$
+
+## Inverse circular functions
+
+Inverse functions: $f(f^{-1}(x)) = x, \quad f(f^{-1}(x)) = x$
+Must be 1:1 to find inverse (reflection in $y=x$
+
+Domain is restricted to make functions 1:1.
+
+### $\arcsin$
+
+$$\sin^{-1}: [-1, 1] \rightarrow \mathbb{R}, \quad \sin^{-1} x = y, \quad \text{where } \sin y = x \text{ and } y \in [{-\pi \over 2}, {\pi \over 2}]$$
+
+### $\arcos$
+
+$$\cos^{-1} \rightarrow \mathbb{R}, \quad \cos^{-1} x = y, \quad \text{where } \cos y = x \text{ and } y \in [0, \pi]$$
+
+### $\arctan$
+
+$$\tan^{-1}: \mathbb{R} \rightarrow \mathbb{R}, \quad \tan^{-1} x = y, \quad \text{where } \tan y = x \text{ and } y \in \left(-{\pi \over 2}, {\pi \over 2}\right)$$